Inequalities

Inequalities

An inequality is a mathematical statement containing two expressions which are not equal. One expression may be less than or greater than the other.

The inequality symbols are:

$<,\ >,\ \leq,\ \geq$

Where:

$<$: less than

$>$: greater than

$\leq$: less than or equal to

$\geq$: greater than or equal to

Linear Inequalities with One Unknown

To solve a linear inequality with one unknown:

1. Collect like terms on one side.
2. Addition or subtraction does not change the inequality direction.
3. Multiplying or dividing by a positive number does not change the direction.
4. Multiplying or dividing by a negative number changes the direction.

Example:

Solve $x - 2 \leq 4$

$x - 2 + 2 \leq 4 + 2$ $x \leq 6$

Final Answer: $x \leq 6$

Linear Inequalities from Practical Situations

Linear inequalities can also come from real-life situations, and they can be represented on a number line.

Important:

Use an empty circle to show an endpoint that is not included in the solution.

Use a solid circle to show an endpoint that is included.

Compound Statement

A compound statement is a mathematical sentence made up of two or more inequalities connected by the words "and" or "or".

Examples:

Using "and":

Solve the compound inequality:

$3 \leq x < 7$

This means that the value of $x$ must satisfy both conditions:

$x \geq 3$ and $x < 7$

Using "or":

Solve the compound inequality:

$x < 2$ or $x \geq 10$

This means the value of $x$ can be less than 2 or greater than or equal to 10.

Example 12

Solve the following compound inequalities and represent the answer on the number line

(a) $10 \leq 2x - 3 < 14$

(b) $7 \leq 3 - 2x < 15$

Solution

(a) Solve $10 \leq 2x - 3 < 14$

Step 1: Add 3 to each part:

$10 + 3 \leq 2x - 3 + 3 < 14 + 3$ $13 \leq 2x < 17$

Step 2: Divide each part by 2:

$\frac{13}{2} \leq x < \frac{17}{2}$ $6\frac{1}{2} \leq x < 8\frac{1}{2}$

(b) Solve $7 \leq 3 - 2x < 15$

Step 1: Subtract 3 from each part:

$7 - 3 \leq 3 - 2x - 3 < 15 - 3$ $4 \leq -2x < 12$

Step 2: Divide each part by -2 and reverse the inequality signs:

$\frac{4}{-2} \geq x > \frac{12}{-2}$ $-2 \geq x > -6$

Rewriting in standard form:

$-6 < x \leq -2$